Calibrate Shifted SABR Model Parameters for Swaption Instrument

This example shows how to calibrate the shifted SABR model parameters for a Swaption instrument when you use a SABR pricing method.

Load Market Data

% Zero curve
ValuationDate = datetime("5-Mar-2016", 'Locale', 'en_US');
ZeroDates = datemnth(ValuationDate,[1 2 3 6 9 12*[1 2 3 4 5 6 7 8 9 10 12]])';
ZeroRates = [-0.33 -0.28 -0.24 -0.12 -0.08 -0.03 0.015 0.028 ...
    0.033 0.042 0.056 0.095 0.194 0.299 0.415 0.525]'/100;
Compounding = 1;
ZeroCurve = ratecurve("zero",ValuationDate,ZeroDates,ZeroRates,'Compounding',Compounding)
ZeroCurve = 
  ratecurve with properties:

                 Type: "zero"
          Compounding: 1
                Basis: 0
                Dates: [16x1 datetime]
                Rates: [16x1 double]
               Settle: 05-Mar-2016
         InterpMethod: "linear"
    ShortExtrapMethod: "next"
     LongExtrapMethod: "previous"

% Define the swaptions
SwaptionSettle = datetime("5-Mar-2016", 'Locale', 'en_US');
SwaptionExerciseDate = datetime("5-Mar-2017", 'Locale', 'en_US');
SwaptionStrikes = (-0.6:0.01:1.6)'/100; % Include negative strikes
SwapMaturity = datetime("5-Mar-2022", 'Locale', 'en_US'); % Maturity of underlying swap
OptSpec = 'call';

Compute Forward Swap Rate by Creating Swap Instrument

Use fininstrument to create a Swap instrument object.

LegRate = [0 0];
Swap = fininstrument("Swap", 'Maturity', SwapMaturity, 'LegRate', LegRate, "LegType",["fixed" "float"],...
    "ProjectionCurve", ZeroCurve, "StartDate", SwaptionExerciseDate)
Swap = 
  Swap with properties:

                     LegRate: [0 0]
                     LegType: ["fixed"    "float"]
                       Reset: [2 2]
                       Basis: [0 0]
                    Notional: 100
          LatestFloatingRate: [NaN NaN]
                 ResetOffset: [0 0]
    DaycountAdjustedCashFlow: [0 0]
             ProjectionCurve: [1x2 ratecurve]
       BusinessDayConvention: ["actual"    "actual"]
                    Holidays: NaT
                EndMonthRule: [1 1]
                   StartDate: 05-Mar-2017
                    Maturity: 05-Mar-2022
                        Name: ""

ForwardValue = parswaprate(Swap,ZeroCurve)
ForwardValue = 7.3271e-04

Load the Market Implied Volatility Data

The market swaption volatilities are quoted in terms of shifted Black volatilities with a 0.8 percent shift.

StrikeGrid = [-0.5; -0.25; -0.125; 0; 0.125; 0.25; 0.5; 1.0; 1.5]/100;
MarketStrikes = ForwardValue + StrikeGrid;
Shift = 0.008;  % 0.8 percent shift
MarketShiftedBlackVolatilities = [21.1; 15.3; 14.0; 14.6; 16.0; 17.7; 19.8; 23.9; 26.2]/100;
ATMShiftedBlackVolatility = MarketShiftedBlackVolatilities(StrikeGrid==0);

Calibrate Shifted SABR Model Parameters

The Beta parameter is predetermined at 0.5. Use volatilities to compute the implied volatility.

Beta = 0.5;

% Calibrate Alpha, Rho, and Nu
objFun = @(X) MarketShiftedBlackVolatilities - volatilities(finpricer("Analytic", 'Model', ...
    finmodel("SABR", 'Alpha', X(1), 'Beta', Beta, 'Rho', X(2), 'Nu', X(3), 'Shift', Shift), ...
    'DiscountCurve', ZeroCurve), SwaptionExerciseDate, ForwardValue, MarketStrikes);

X = lsqnonlin(objFun, [0.5 0 0.5], [0 -1 0], [Inf 1 Inf]);
Local minimum possible.

lsqnonlin stopped because the final change in the sum of squares relative to 
its initial value is less than the value of the function tolerance.
Alpha = X(1);
Rho = X(2);
Nu = X(3);

Create SABR Model Using the Calibrated Parameters

Use finmodel to create a SABR model object.

SABRModel = finmodel("SABR",'Alpha',Alpha,'Beta',Beta,'Rho',Rho,'Nu',Nu,'Shift',Shift)
SABRModel = 
  SABR with properties:

             Alpha: 0.0135
              Beta: 0.5000
               Rho: 0.4654
                Nu: 0.4957
             Shift: 0.0080
    VolatilityType: "black"

Create SABR Pricer Using Calibrated SABR Model and Compute Volatilities

Use finpricer to create a SABR pricer object and use the ratecurve object for the 'DiscountCurve' name-value pair argument.

SABRPricer = finpricer("Analytic", 'Model', SABRModel, 'DiscountCurve', ZeroCurve)
SABRPricer = 
  SABR with properties:

    DiscountCurve: [1x1 ratecurve]
            Model: [1x1 finmodel.SABR]

SABRShiftedBlackVolatilities = volatilities(SABRPricer, SwaptionExerciseDate, ForwardValue, SwaptionStrikes)
SABRShiftedBlackVolatilities = 221×1

    0.2978
    0.2911
    0.2848
    0.2787
    0.2729
    0.2673
    0.2620
    0.2568
    0.2518
    0.2470
      ⋮

figure;
plot(MarketStrikes, MarketShiftedBlackVolatilities, 'o', ...
    SwaptionStrikes, SABRShiftedBlackVolatilities);
h = gca;
line([0,0],[min(h.YLim),max(h.YLim)],'LineStyle','--');
ylim([0.13 0.31])
xlabel('Strike');
legend('Market quotes','Shifted SABR', 'location', 'southeast');
title (['Shifted Black Volatility (',num2str(Shift*100),' percent shift)']);

Price Swaption Instruments Using Calibrated SABR Model and SABR Pricer

% Create swaption instruments
NumInst = length(SwaptionStrikes);
Swaptions(NumInst, 1) = fininstrument("Swaption", ...
    'Strike', SwaptionStrikes(1), 'ExerciseDate', SwaptionExerciseDate(1), 'Swap', Swap);
for k = 1:NumInst
    Swaptions(k) = fininstrument("Swaption", 'Strike', SwaptionStrikes(k), ...
        'ExerciseDate', SwaptionExerciseDate, 'Swap', Swap, 'OptionType', OptSpec);
end
Swaptions
Swaptions=221×1 object
  16x1 Swaption array with properties:

    OptionType
    ExerciseStyle
    ExerciseDate
    Strike
    Swap
    Name
      ⋮

% Price swaptions using the SABR pricer
SwaptionPrices = price(SABRPricer,Swaptions);

figure;
plot(SwaptionStrikes, SwaptionPrices, 'r');
h = gca;
line([0,0],[min(h.YLim),max(h.YLim)],'LineStyle','--');
xlabel('Strike');
title ('Swaption Price');